Methods for computing the CRC of a message from the incremental CRCs of composite sub-messages

ABSTRACT

A method of generating a CRC for a composite sub-message based on a CRC generating polynomial having at least two factors. The composite sub-message includes sub-message data and a number, n, of trailing zeros. The method includes generating a first remainder based on the sub-message data and a first factor of the CRC generating polynomial. A second remainder is generated based on the sub-message data and a second factor of the CRC generating polynomial. The CRC for the composite sub-message is generated based on adjusted versions of the first and the second remainders.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is a continuation-in-part of U.S. patent application Ser. No. 10/080,886, filed Feb. 22, 2002 and entitled “Methods for Computing the CRC of a Message from the Incremental CRCs of Composite Sub-Messages”.

FIELD OF THE INVENTION

[0002] This invention pertains generally to error detection and more particularly to a method of accumulating cyclic redundancy checks of sub-messages regardless of arrival order.

BACKGROUND

[0003] Coding systems using cyclic redundancy check (CRC) techniques are readily implemented yet provide powerful error detection capabilities. Accordingly, CRC techniques are widely used in, for example, disk controllers, Internet protocols such as such as IP and iSCSI, and other networking protocols including ethernet. In the CRC technique, a block of d data bits denoted as a frame is joined with an extra block of m bits called the frame check sequence (FCS). Just like a checksum such as a parity bit, the FCS introduces redundancy into the transmitted (d+m) bit codeword that permits the receiver to detect errors. All the bits are treated as binary coefficients of polynomials. A receiver will detect errors in the received (d+m) bit codeword by dividing (using polynomial arithmetic) the codeword with a generator polynomial. If the remainder from this division is zero, a CRC-enabled receiver will assume that the transmitted codeword contains no errors.

[0004] As discussed above, certain Internet protocols require CRC coding to provide error detection. In these protocols, a data message may be packetized or divided into sub-messages for transmission. For example, an iSCSI data message may be protected with its CRC FCS and transmitted via multiple IP packets (which may be denoted as sub-messages) that may arrive in any order. The CRC coding of the FCS, however, is based on the original data message and not the IP packets/sub-messages. Conventionally, a receiver may perform a CRC calculation on the resulting sub-messages in one of two approaches. In a first approach to perform the CRC division/calculation, a conventional receiver could accumulate the sub-messages to reconstruct the message and divide the message by the generator polynomial. If the remainder from this division is zero, the message is assumed to be error free. Because the CRC division is performed after all the sub-messages have been received, there is extra latency causing undesirable delay. In addition, the receiver must have read access to the memory storing the accumulated sub-messages. Even if such memory access is practical, the extra loading on the memory bus further impacts system performance.

[0005] Alternatively, in a second approach to perform the CRC calculation, a receiver could compute the CRC remainder by performing a CRC division on each sub-message as it arrives in order using a CRC computation engine. In sequence processing of the sub-messages is required to ensure that the CRC computation engine has the proper initial state. The processed sub-messages could then be delivered to a remote memory not accessible to the CRC computation engine, eliminating the loading on the memory bus suffered by the previous approach. However, because the second approach requires in sequence delivery of the sub-messages, it cannot be applied where support of out of order sub-message delivery is required or desired.

[0006] Accordingly, there is a need in the art for a CRC computation technique that calculates the CRC of sub-messages regardless of arrival order.

SUMMARY

[0007] One form of the present invention provides a method of generating a CRC for a composite sub-message based on a CRC generating polynomial having at least two factors. The composite sub-message includes sub-message data and a number, n, of trailing zeros. The method includes generating a first remainder based on the sub-message data and a first factor of the CRC generating polynomial. A second remainder is generated based on the sub-message data and a second factor of the CRC generating polynomial. The CRC for the composite sub-message is generated based on adjusted versions of the first and the second remainders.

BRIEF DESCRIPTION OF THE DRAWINGS

[0008] The various aspects and features of the present invention may be better understood by examining the following figures, in which:

[0009]FIG. 1 is a graphical representation of a message divided into sub-messages and a composite sub-message corresponding to one of the sub-messages.

[0010]FIG. 2 is a logical representation of a circuit for performing the field squaring of a 4-tuple binary number according to a generating polynomial.

[0011]FIG. 3 illustrates a feedback shift register for performing the field squaring shown in FIG. 2.

[0012]FIG. 4 illustrates a feedback shift register used as a CRC computation engine.

[0013]FIG. 5 illustrates the feedback shift register of FIG. 4 during the stage when it processes the n trailing zeroes.

[0014]FIG. 6 is a flow chart for a method of adjusting a CRC according to one embodiment of the invention.

[0015]FIG. 7 illustrates a feedback shift register for simultaneously field squaring and stepping a state according to one embodiment of the invention.

[0016]FIG. 8 illustrates a state machine having four modes of operation for adjusting a CRC according to one embodiment of the invention.

[0017]FIG. 9a is a flow chart for a method of adjusting a CRC according to one embodiment of the invention.

[0018]FIG. 9b is a flow chart for a table tookup technique for the method of FIG. 9a according to one embodiment of the invention.

[0019]FIG. 10 is a data flow diagram illustrating a first method for generating an incremental CRC based on a composite CRC generating polynomial according to one embodiment of the present invention.

[0020]FIG. 11 is a data flow diagram illustrating a second method for generating an incremental CRC based on a composite CRC generating polynomial according to one embodiment of the present invention.

[0021]FIG. 12 is a schematic diagram illustrating a circuit for implementing a mapping from individual remainders to a composite remainder for an example CRC generating polynomial according to one embodiment of the present invention.

DETAILED DESCRIPTION

[0022] In the following detailed description of the preferred embodiments, reference is made to the accompanying drawings, which form a part hereof, and in which is shown by way of illustration specific embodiments in which the invention may be practiced. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present invention. The following detailed description, therefore, is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims.

[0023]FIG. 1 illustrates a (d+m) bit binary message 10. Message 10 may be arbitrarily divided into chunks of bits denoted as sub-messages 12, 14, and 16. Message 10 can be expressed as the modulo-2 summation of the sub-messages. However, each sub-message must be modified with zeroes in place of the data from the remaining sub-messages. For example, consider sub-message 14. To be used in a modulo-2 summation to form message 10, sub-message 14 must be modeled as composite sub-message 18. As can be seen, composite sub-message 18 has zeroes 20 in place of the sub-message data from sub-messages 12 and 16. When modeled in this fashion, each composite sub-message will have the same length (number of bits) as the original message 10.

[0024] Because the CRC calculation is a linear transformation, the CRC of message 10 is a sum of the CRC's of the composite sub-messages. If a conventional serial CRC computation engine is used to calculate the CRC of a composite sub-message, the leading zeroes may be ignored if the initial state of the CRC computation engine is zero. In such a case, to begin calculation of the CRC for a composite sub-message, the computation engine may begin by calculating the CRC of the corresponding sub-message. However, the resulting CRC of the sub-message must be adjusted to form the CRC of the composite sub-message. As used herein, the CRC of a composite sub-message will be denoted as an incremental CRC. The required adjustment may be made by continuing to cycle the computation engine with no further input for as many clock cycles n as there are trailing zeroes.

[0025] In a first embodiment of the invention, which will be denoted the “optimal method,” the CRC of a sub-message corresponding to a composite sub-message having n trailing zeroes may be adjusted in log n clock cycles to form the incremental CRC. These incremental CRCs can then be modulo-2 summed (equivalent to exclusive ORing) in any order to form the CRC for the original message.

[0026] To understand the optimal method, certain coding concepts that are well known to one of ordinary skill in the art must be appreciated. For example, a field is a set of elements upon which two binary operations are defined. These binary operations may be referred to as “addition” and “multiplication.” However, to prevent confusion with the usual binary operations of addition and multiplication, these field binary operations may be referred to as “field addition” and “field multiplication” using the symbols “+” and “*,” respectively. In addition to other requirements, the result of either binary operation between two field elements must be equal to another element within the field. A field with a finite number of elements is of particular interest and is referred to as a “Galois field.”

[0027] The simplest Galois field is the set of binary numbers {1, 0}, which is denoted as GF(2). In this case, the field addition and field multiplication binary operations are the familiar modulo-2 addition and modulo-2 multiplication. Using GF(2), coding theorists may construct extension Galois fields having 2^(m) elements denoted as GF(2^(m)). To begin such a construction, consider a polynomial with a single variable x whose coefficients are from GF(2). As used herein, “polynomial” will refer only to polynomials having coefficients from GF(2). A polynomial with one variable x whose largest power of x with a nonzero coefficient is m is said to be of degree m. For example the polynomial P(x)=x⁴+x+1 is of degree 4. As will be explained further herein, two types of polynomials are of particular importance for the optimal method: irreducible and primitive. An irreducible polynomial P(x) of degree m is not divisible by any polynomial of degree smaller than m but greater than zero. A primitive polynomial P(x) is an irreducible polynomial wherein the smallest possible integer n in which P(x) divides the polynomial x^(n)+1 is n=2^(m)−1.

[0028] A primitive polynomial of degree m can be used to generate the elements of a Galois field GF(2^(m)). For example, consider the primitive polynomial P(x)=1+x+x⁴. The degree of this polynomial is 4 so it may be represented by a 4 bit binary number (4-tuple representation). In general, a primitive polynomial of degree m can be used to generate 2^(m)−1 unique states (ignoring the trivial all-zero state). Each state may have either a polynomial or m-tuple representation. It is customary to use the element a instead of x in the following discussion. Each polynomial may also be represented by a power of α (power representation) as will be explained further herein. The four simplest polynomials would simply be 1, α, α², and α³, respectively. These polynomials would correspond to a 4-tuple form (most significant bit first) as 0001, 0010, 0100, and 1000, respectively. The power representation corresponds directly to α⁰, α¹, α², and α³, respectively. Further powers of α may be developed by equaling the generating polynomial P(α)=1+α+α⁴ to zero. This leads to the recursive formula α⁴=α+1. By continuing to multiply by α and apply the recursive formula, the powers of a may be derived as given by the following Table 1: TABLE 1 Binary (4-tuple Power Representation Polynomial Representation Representation) 0 0 0000 α⁰  α⁰ 0001 α¹  α 0010 α²  α² 0100 α³  α³ 1000 α⁴  α + α⁰ 0011 α⁵  α² + α 0110 α⁶  α³ + α² 1100 α⁷  α³ + α + α⁰ 1011 α⁸  α² + α⁰ 0101 α⁹  α³ + α 1010 α¹⁰ α² + α + α⁰ 0111 α¹¹ α³ + α² + + α 1110 α¹² α³ + α² + α + α⁰ 1111 α¹³ α³ + α² + α⁰ 1101 α¹⁴ α³ + α⁰ 1001

[0029] Examination of Table 1 shows that the various elements of this Galois field may be considered as states in a finite state machine. For example, if the state machine is in state 0110 (α⁵), the next state would be 1100 (α⁶). For a given state, the next state is achieved by multiplying by α.

[0030] Another concept important to the invention is that of finite field squaring an element within the Galois field GF(2^(m)) generated by a polynomial P(x). As an example, consider again the generating (and primitive) polynomial P(α)=1+α+α⁴. If one of the polynomial elements in the resulting Galois field GF(2^(m)) (represented as b₃α³+b₂α²+b₁α+b₀) is finite field squared, it can be shown that the result is b₃α⁶+b₂α⁴+b₁α²+b₀. By using the recursion relationship α⁴=α+1, the result becomes b₃α³+(b₁+b₃)α²+b2α+(b₀+b₂). Thus, given an arbitrary 4-tuple element [b₃, b₂, b₁, b₀] from Table 1, it will be finite field squared by the circuit 24 shown in FIG. 2 to produce the 4-tuple element [c₃, c₂, c₁, c₀]. Because addition modulo-2 is equivalent to exclusive ORing, the modulo-2 addition of coefficients b₁, b₃, b₀, and b₂ is performed by XOR gates 26. Note that, in general, for a generating polynomial P(x) of degree m, finite field squaring an m-tuple binary number is equivalent to multiplying modulo P(x) the m-tuple binary number by itself.

[0031] Turning now to FIG. 3, a shift register 30 is shown for implementing the field squaring of a 4-tuple binary number, wherein the field is generated by the polynomial P(x)=x⁴+x+1. Each stage 31 is a D-type flip-flop. Stages 31 store the 4 bits of the resulting 4-tuple binary number. Because this 4-tuple number may be construed as a polynomial, stages 31 are labeled accordingly from the term x⁰ to the term x³. The contents of each stage 31 may then be construed as the coefficient of the corresponding polynomial factor. These coefficients may be denoted as a 4-bit vector Y(n)=[y₀, y₁, y₂, y₃], where y₃ through y₀ correspond to the contents of stages 31 x³ through x⁰, respectively. Vector Y(n) may also be considered as the state of shift register 30. Each time stages 31 are clocked, the 4-tuple binary number is field squared. XOR gates 26 provide the same function as in FIG. 2. Comparing shift register 30 of FIG. 3 to the squaring transformation illustrated in FIG. 2 illustrates the identity. For example, y₀(n+1)=y₀(n){circumflex over ( )}y₂(n), where {circumflex over ( )} represents the XOR operation. The next state Y(n+1) of register 30 is derived from Y(n). This next state is given by the expression Y(n+1)=S·Y(n), where S is the squaring matrix given by the following Equation 1: $\begin{matrix} {S = \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix}} & {{Equation}\quad 1} \end{matrix}$

[0032] Each row of the squaring matrix may be derived from examination of shift register 30. For example, from shift register 30 it can be seen that y₁(n+1)=y₂(n). Thus, y₁(n+1)=[0010]·[y₀(n)y₁(n)y₂(n)y₃(n)]^(T), where T stands for the transpose operation.

[0033] With these concepts in mind, the optimal method of the present invention may be discussed further. This method uses a conventional CRC computation engine to calculate the incremental CRC of a composite sub-message. As is well known, such CRC computation engines may be implemented in either hardware or software, in either a serial or parallel configuration. Turning now to FIG. 4, a hardware implementation is illustrated using a feedback shift register 32 to calculate the CRC using the generating polynomial P(x)=x⁴+x+1. Each stage 31 is a D-type flip-flop. As discussed with respect to FIG. 3, the contents of each stage (or equivalently, the state of register 32) corresponds to a vector Y(n). Because this is the same generating polynomial as discussed above, the bits stored in the stages 31 of shift register 32 will correspond to the 4-tuple representation discussed with respect to the Table 1. The bitstream carrying the sub-message is clocked in one bit at a time into the shift register 32 at the input, d_(in). An XOR gate 36 receives this input and the Q output of stage 31 for bit y₃(n) to form the D input for stage 31 holding bit y₀(n). Similarly, another XOR gate 36 receives the Q output of stage 31 for bit y₀(n) as well as the Q output of stage 31 for bit y₃(n) to form the D input for stage 31 for bit y₁(n).

[0034] During the CRC calculation stage, a switch 33 couples to feedback path 34 to maintain the required feedback. At each clock cycle, the 4-tuple binary number represented by the contents of the stages 31 advances to the next state of the Galois field defined by the generating polynomial P(x), as discussed with respect to Table 1. Note that with respect to the received sub-message, the composite sub-message has n trailing zeroes as discussed with respect to FIG. 1. Should the shift register 32 be used without implementing the CRC computation technique of the present invention, switch 33 would be maintained in this position for these n additional clock cycles although no further data would enter at input d_(in). After these n clock cycles, switch 33 couples to output d_(out) so that the CRC may be read out.

[0035] Because switch 33 remains coupled to feedback path 34 and no data enters input d_(in) during this time, feedback shift register 32 may be represented by the feedback shift register 40 of FIG. 5 during these n clock cycles. Here, stages 31 store the 4 bits y₃(n) through y₀(n) defining vector Y(n) as discussed for shift register 30 of FIG. 4. Because the input d_(in) of FIG. 4 is always zero as the n trailing zeroes are processed, the XOR gate 36 at the input to stage 31 holding bit y₀(n) is absent. The bits y₃(n) through y₀(n) stored in stages 31 represent one of the elements of the Galois field generated by P(x) as set forth in Table 1. By defining y₃(n) through y₀(n), shift register 40 forms a finite state machine whose present state (corresponding to the current clock cycle and denoted as the 4-bit vector Y(n)) is given by the current value of these bits. The next state Y(n+1) of shift register 40 (corresponding to the next clock cycle) is computed from the present state Y(n) by the linear relationship Y(n+1)=A·Y(n), where A is the state transition matrix and arithmetic is done modulo-2 (i.e, addition is an XOR operation). The columns of state transition matrix A are the binary representation of the states (in their power representation): α, α², α³, and α⁴. Thus, the state transition matrix A for this particular generating polynomial is given by the following Equation 2: $\begin{matrix} {A = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}} & {{Equation}\quad 2} \end{matrix}$

[0036] Note that once the sub-message has been processed in shift register 32 of FIG. 4 (i.e., all the bits of the sub-message have been clocked into this register) shift register 32 must advance n states to adjust the CRC so that an incremental CRC for the corresponding composite sub-message may be determined. Before this adjustment, stages 31 of shift register 32 store the unadjusted CRC for the sub-message. Rather than wait through n clock cycles, the following steps of the optimal method will adjust the CRC of the sub-message in as little as log(n) clock cycles.

[0037] Because shift register 32 has only 15 states, even if the number n of trailing zeroes exceeds 15, the actual number of states N that shift register 32 must be advanced is n modulo 15. Thus, n may be expressed as a 4-tuple number N, where N=modulo (2^(m)−1) of n (or n mod (2^(m)−1)). As a first step, each bit of N is examined, starting with the most significant bit. For each examined bit of N, whether 0 or 1, the contents of the shift register 32 are field squared using the shift register 30 discussed with respect to FIG. 3. In addition, for each examined bit that equals 1, the contents of the shift register 32 are advanced one state. When all bits of N have been examined, the CRC of the sub-message will have been adjusted to form the incremental CRC of the corresponding composite sub-message.

[0038] A flowchart for the optimal method is illustrated in FIG. 6. As a first step in the method, the sub-message m-bit CRC is calculated and stored in an m-bit memory location at step 41. The corresponding composite sub-message will have n trailing zeroes. At step 42, each bit of N=n mod (2^(m)−1) is examined in order from the most significant bit to the least significant bit. For each examined bit of N at step 43, the contents of the m-bit memory location are field squared. Finally, if the examined bit of N equals one at step 44, the contents of the m-bit memory location are advanced to the next state as determined by the Galois field defined by the CRC generating polynomial. Should bits of N remain unexamined, steps 43 and 44 repeat until all bits are examined.

[0039] Note that the preceding method may take up to two clock cycles for each bit of N (one clock cycle for the field squaring and potentially one more clock cycle for the advancing step should the examined bit of N equal one). By pre-combining the squaring and stepping matrices S and A, it is possible to perform both transformations (squaring and stepping) in a single clock cycle. The one state advance must be performed after the squaring such that a new matrix B=A·S is required. Given an m-tuple CRC at clock cycle n represented by the vector Y(n), the m-tuple CRC at clock cycle n+1 represented by Y(n+1) would be given by the linear relationship Y(n+1)=B·Y(n). With the matrices A and S being as described above, the B matrix for the generating polynomial P(x)=x⁴+x+1 is given by the following Equation 3: $\begin{matrix} {B = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}} & {{Equation}\quad 3} \end{matrix}$

[0040] Turning now to FIG. 7, a shift register 45 for performing the combined field squaring and stepping in one clock cycle is illustrated. Stages 31 store the CRC bits, which will be adjusted in one clock cycle. As discussed with respect to FIG. 5, the stored bits may be considered to form a vector Y(n)=[y₀(n)y₁(n)y₂(n)y₃(n)]. A comparison of shift register 45 with matrix B illustrates how shift register 45 performs the combined field squaring and stepping. For example, matrix B indicates that bit y₁(n+1) equals the modulo-2 addition of bits y₀(n), y₂(n), and y₃(n). Thus, shift register 45 has an XOR gate 47 at the input to stage 31 labeled x¹, where XOR gate 47 receives the outputs from stages 31 storing bits y₀(n), y₂(n), and y₃(n).

[0041] Turning now to FIG. 8, a state machine 100 for performing an m-bit CRC adjustment will thus have four modes of operation determined by bits M₁ and M₀. The CRC generating polynomial for state machine 100 is as discussed with respect to FIGS. 2 through 5. Although this generating polynomial is used as an example, it will be appreciated that state machine 100 may be generalized to correspond to whatever generating polynomial is desired. D-type flip-flops 31 store the CRC bits corresponding to the polynomial representation x³ through x⁰. These bits are defined with respect to clock cycles controlling each register 31. Thus, the Q output of a register 31 will correspond to the nth clock cycle whereas the D input of a register 31 will correspond to the Q output at the (n+1)th clock cycle. Bits M₁ and M₀ control 4:1 multiplexers 110. Each 4:1 multiplexer provides the D input for its register 31.

[0042] Each multiplexer 110 has four inputs, numbered from 0 to 3. If M₁=0 and M₀=0, multiplexers 110 will select for the zeroth input. If M₁=0 and M₀=1, multiplexers 110 will select for the first input. If M₁=1 and M₀=0, multiplexers 110 will select for the second input. Finally, if M₁=1 and M₀=1, multiplexers 110 will select for the third input. Accordingly, the modes are as follows:

[0043] In a first mode of state machine 100, it will initialize by receiving the CRC of a sub-message, represented by bits I₃, I₂, I₁, and I₀ in order from most significant bit to the least significant bit. By setting bits M₁ and M₀ to zero and zero, respectively, this first mode is selected for by multiplexers 110. In a second mode, it will field-square the m bits stored by the state machine that represent the current state. By setting bits M₁ and M₀ to zero and one, respectively, this second mode is selected for by multiplexers 110. In a third mode, the state machine will advance its stored bits representing the current state to the next state. By setting bits M₁ and M₀ to one and zero, respectively, this third mode is selected for by multiplexers 110. Finally, in a fourth mode, the state machine will both field square and advance to the next state its stored bits representing the current state. By setting bits M₁ and M₀ to one and one, respectively, this fourth mode is selected for by multiplexers 110. Advantageously, the complexity of the field squaring operation (which is basically a field multiplier with both operands the same) is equivalent to the stepping transformation. This equivalent complexity is a result of the modulo P(x) and modulo 2 arithmetic. In contrast, a general binary multiplier is much more complex.

[0044] Note that the optimal method for adjusting a CRC is similar to a prior art algorithm of performing exponentiation of integers using a single accumulator. This algorithm is as follows. Suppose we want to compute β^(n). The accumulator is initialized with 1 and the binary representation of the exponent n is inspected, in order from most significant bit (msb) first. For every bit of n, the current contents of the accumulator are squared. For every bit of n that equals 1, the accumulator is multiplied by β after the squaring. After inspecting all bits of n, the accumulator will hold β^(n). For example, suppose we want to raise β to the power n=5. Table 2 below shows how the accumulator is initialized to 1 and how its contents are transformed at each step as the bits of n are inspected. After the last step, the accumulator contains β⁵(1¹⁶*β⁴*β¹) as expected. The reason the algorithm works is that, if we decompose the exponent into its powers of two, we can write the desired result, β⁵, as β⁴+β¹ and, by introducing β as a new factor whenever n has a 1, that β factor (and all others) will undergo the correct number of squarings. TABLE 2 Contents of the Accumulator in terms of After n factors clock # 1 0 0 (msb) 1² 1 1 1⁴ β 2 0 1⁸ β² 3 1 (lsb) 1¹⁶ β⁴ β 4

[0045] Note that this algorithm works only if the initial accumulator state is 1. As such, it is not directly applicable to the problem of adjusting a CRC, which will start from an arbitrary state.

[0046] As described above, a primitive polynomial of degree m has 2^(m)−1 unique states (excluding the trivial case of all zeroes). Given a current state of a CRC computation engine programmed with a primitive polynomial, the next state is uniquely defined. Accordingly, such a CRC engine may be advanced to the next state during implementation of the optimal method. However, CRC computation engines programmed with an irreducible polynomial may also implement the optimal method of the present invention. For example, consider the irreducible polynomial given by P(x)=x⁴+x³+x²+x+1. This polynomial will have the recursion relationship: x⁴=x³+x²+x+1, which may be used to derive all of its states as given by the following Table 3. TABLE 3 Binary (one of several possible 4- tuple represent- Power Representation Polynomial Representation ations) α⁰  x⁰ 0001 α¹  x¹ 0010 α²  x² 0100 α³  x³ 1000 α⁴  x³ + x² + x¹ +x⁰ 1111 α⁵  x⁰ 0001 α⁶  x¹ 0010 α⁷  x² 0100 α⁸  x³ 1000 α⁹  x³ + x² + x¹ +x⁰ 1111 α¹⁰ x⁰ 0001 α¹¹ x¹ 0010 α¹² x² 0100 α¹³ x³ 1000 α¹⁴ x³ + x² + x¹ +x⁰ 1111

[0047] Note that instead of 15 unique states (compared to a primitive polynomial of degree 4), there are only 5 unique states. However, because the number of unique states is a factor of 15, a CRC computation engine programmed with such an irreducible polynomial may be advanced to the next state as required by the optimal method. For example, consider the state given by the 4-tuple representation [1000]. Regardless of whether we designate this state as α³, α⁸, or α¹³, the next state is uniquely given by [1111].

[0048] In the more general case, where the CRC generating polynomial is neither primitive nor irreducible, the optimal method as described above will not work, because for certain states, the following state will not be uniquely determined. Because a CRC computation engine could not be advanced to the next state for these states, the optimal method as described above would break down. However, the optimal method will work even in this general case if the initial starting state of the CRC computation engine is equal or congruent to one. In addition, the optimal method will work in the general case by separating the CRC generating polynomial into factors, as will be described below with reference to FIGS. 10-12.

[0049] An alternate embodiment of the invention provides a method for CRC adjustment that will always work regardless of the form of the generating polynomial (i.e., regardless of whether the generating polynomial is primitive or irreducible). As used herein, this method will be referred to as the “general method.” Compared to the optimal method, the general method is slower. However, the general method is much faster at adjusting a CRC than present known methods and shares most of the advantages of the optimal method.

[0050] The general method uses a well-known relationship from modular arithmetic that (x*y) mod m=(x mod m*y mod m) mod m where x, y and m are integers. This relationship also holds when x, y and m are polynomials. Computing the CRC of a message polynomial can be described mathematically as computing the residue of the message polynomial modulo the CRC polynomial. For simplicity we will ignore the fact that normally the message is pre-multiplied by x^(m) to make room for the m-bit CRC.

[0051] A composite sub-message polynomial can be expressed as the product of the sub-message polynomial and x^(n) (the polynomial representing the bit stream of 1 followed by n zeroes) where n is the number of trailing zeroes. The above-mentioned mathematical relationship from modular arithmetic may be used to compute the adjusted (incremental) CRC or [x^(n)·A(x)]mod P(x) where n is the number of trailing zeroes, P(x) is the generating polynomial, and A(x) is the sub-message polynomial. Using the above relationship we can say that the above is equal to [x^(n)mod P(x)·A(x)mod P(x)]mod P(x).

[0052] Turning now to FIG. 9a, a flowchart summarizing the general method is illustrated. At step 50, the general method computes A(x) mod P(x), the CRC of a sub-message without trailing zeroes (i.e. without regard to position). At step 51, compute x^(n) mod P(x) using a single lookup table. This step could be done in parallel with step 1. Finally, at step 52, combine the results from steps 1 and 2 by field multiplying together and field dividing by P(x). The remainder produces the adjusted CRC.

[0053] The table lookup performed in step 51 may be performed by factoring x^(n) into powers of two and then multiplying. For example, x²⁷=x¹⁶·x⁸·x²·x¹, resulting in an initial table lookup for x¹⁶mod P(x), x⁸mod P(x), x²mod P(x), and x mod P(x) and then multiplying together the looked up results. Turning now to FIG. 9b, this specific table lookup method is illustrated. In step 51 a, n is factored into powers of 2. In step 51 b, each factor is table-looked up and then multiplied together. This factoring of n minimizes the size of the required lookup table. Note that when n is smaller than the degree of P(x) there is no need for the table lookup and, because the polynomial has a single term, multiplication consists of simply shifting left.

[0054] The general method may use a circuit that will field multiply two short (same size as the CRC) polynomials together and will simultaneously field divide by P(x) since we just want the remainder. The field division by P(x) may be performed by a standard CRC computation engine. However, the multiplication requires more computation power than the optimal method for large CRCs. Also, the general method requires a lookup table. Should the exponent n be factored into powers of 2, then processing message sizes of 2 n bits with the general method requires a look-up table with n entries. Each entry is the same size as the CRC. For instance, a message size of 16 K bytes with a 32 bit CRC would require a table of at most 17 32-bit entries.

[0055] Regardless of whether the optimal method or the general method is implemented, the present invention provides faster CRC computation than that provided by the prior art. Moreover, present CRC computation techniques implemented in software use a hardware offload computation engine that receives the entire message. Using the present invention, the message could be divided into several sub-messages, where each sub-message is processed by its own CRC computation engine working in parallel with the other engines. The adjusted CRCs of the sub-messages could then be put together using either hardware or software to compute the message CRC. In this fashion, the CRC computation could be parallelised.

[0056] In addition, pre-computation of fixed fields or the invariant parts of a message could be implemented to speed CRC computation time with the present invention. For example, a large binary word may be hashed into a smaller word of fixed size that can be used to index a table directly. In a hash table lookup, a large binary word is related to an entry in a table but the size of the binary word is too large for the binary word to be used directly as an index to the table. The present invention permits pre-computing part of the hash if some portion of the large binary word is known beforehand.

[0057] The invention may also be applied in routing or switching applications, when changing address or other fields in a message and recomputing a CRC. When a router or switch changes values of some field in a packet, adjusting the CRC using the present invention takes significantly less time than computing a new CRC for the whole message. Furthermore, by computing an adjustment to the CRC rather than computing a new CRC, the packet continues to be protected against errors while in the router or switch. If a new CRC was computed and an error occurred in the switch/router between checking the original CRC and computing the new CRC, the packet would have a good CRC despite having been corrupted by an error. If the invention is used with respect to changes to fields in a packet, the CRC adjustment is calculated based on the bits that have been changed. In other words, the CRC adjustment is based on the XOR of the new and old field values. Alternatively, a CRC adjustment is subtracted from the old field values and a CRC adjustment is added for the new field values.

[0058] Other applications of the invention include signature analysis for hardware fault detection, computation of syndromes in error correction techniques, and quickly skipping over n states of a circuit that generates some sequence of all elements from a Galois field. In addition, the invention may be used to determine if a polynomial is primitive. Conventional techniques for determining whether a polynomial is primitive involve initializing a CRC register to a value and then cycling through all states to determine if the polynomial is maximal length. The number of operations of such a technique is proportional to the cycle length, which grows quickly with polynomial order. In contrast, should the calculation be performed according to the present invention, the number of operations would be proportional to the log of the cycle length, which grows much more slowly with the cycle length.

[0059] Most practical CRC generating polynomials in use today are either primitive polynomials (e.g., the Ethernet polynomial) or composite polynomials (e.g., the iSCSI polynomial) that are composed of the product of a primitive polynomial and the polynomial x+1. A composite polynomial P(x) of degree m is divisible by one or more polynomials of degree smaller than m but greater than zero. Composite polynomials do not meet the restrictions set forth above for the optimal method. However, one embodiment of the present invention provides a technique that allows the optimal method to be applied to composite polynomials, including the iSCSI polynomial.

[0060] One form of the present invention applies to composite polynomials whose factors are relatively prime. Most practical polynomials are either primitive, or the product of two polynomials that have no common factors (i.e., that are relatively prime). In fact, one factor is usually primitive and the other is usually x+1 (also primitive), and thus the factors are relatively prime. For such composite polynomials, the Chinese Remainder Theorem for Polynomials guarantees that there will be a unique one-to-one mapping between the composite remainder (i.e., the CRC computed using the composite polynomial) and the individual remainders (the CRCs computed using the factors of the composite polynomial). One form of the present invention provides a method for determining the mapping from the individual remainders to the composite remainder.

[0061]FIG. 10 is a data flow diagram illustrating a first method 200 for generating an incremental CRC based on a composite CRC generating polynomial according to one embodiment of the present invention. FIG. 11 is a data flow diagram illustrating a second method 300 for generating an incremental CRC based on a composite CRC generating polynomial according to one embodiment of the present invention. Before describing FIGS. 10 and 11, some symbols used in these Figures will be defined:

[0062] 1. A(x) is a polynomial representing a sub-message (i.e., sub-message data without trailing zeros);

[0063] 2. P(x) is a composite CRC generating polynomial;

[0064] 3. P1(x) is a first factor of P(x);

[0065] 4. P2(x) is a second factor of P(x) (note that if x+1 is one of the factors of P(x), the x+1 factor is made P2(x)); and

[0066] 5. Rp(x) is a composite partial (unadjusted) remainder or CRC (i.e., Rp(x)=A(x) mod P(x)).

[0067] If the CRC generating polynomial, P(x), were primitive, the unadjusted CRC, Rp(x), of the sub-message, A(x), could be computed by dividing the sub-message, A(x), by the generating polynomial, P(x). And the unadjusted CRC, Rp(x), could be adjusted using the optimal method described above to obtain the adjusted or incremental CRC, R(x). For the case where the CRC generating polynomial, P(x), is not primitive, but is a composite of at least two factors, P1(x) and P2(x), the optimal method can still be used, as will be described below with reference to FIGS. 10 and 11.

[0068] As shown in FIG. 10, in step 204 of the first method 200, the sub-message 202 (A(x)) is divided by the first factor, P1(x), to obtain a first factor partial remainder 206 (R1p(x)). The division is step 204 is represented by the following Equation 4:

R1p(x)=A(x) mod P1(x)  Equation 4

[0069] During the same pass through the message, in step 212, the sub-message 202 (A(x)) is divided by the second factor, P2(x), to obtain a second factor partial remainder 214 (R2p(x)). The division in step 212 is represented by the following Equation 5:

R2p(x)=A(x) mod P2(x)  Equation 5

[0070] In step 208, the first factor partial remainder 206 (R1p(x)) is adjusted based on the number, n, of trailing zeros in the composite sub-message to generate an adjusted first factor remainder 210 (R1(x)). The adjustment in step 208 is represented by the following Equation 6:

R1(x)=R1p(x)x^(n)mod P1(x)=Rp(x)x^(n)mod P1(x)  Equation 6

[0071] Likewise, in step 216, the second factor partial remainder 214 (R2p(x)) is adjusted based on the number, n, of trailing zeros in the composite sub-message to generate an adjusted second factor remainder 218 (R2(x)). The adjustment in step 216 is represented by the following Equation 7:

R2(x)=R2p(x)x^(n)mod P2(x)=Rp(x)x^(n)mod P2(x)  Equation 7

[0072] In one embodiment, the adjustment performed in steps 208 and 216 is accelerated by using the optimal method described above, since the factors P1(x) and P2(x) are irreducible.

[0073] In step 220, an adjusted composite remainder or incremental CRC 222 (R(x)), which is an adjusted version of Rp(x), is obtained from the individual remainders R1(x) and R2(x) by a mapping described in further detail below.

[0074]FIG. 11 is a data flow diagram illustrating a second method 300 for generating an incremental CRC based on a composite CRC generating polynomial according to one embodiment of the present invention. In step 304, a composite partial remainder 306 (Rp(x)) of the sub-message 302 (A(x)) is calculated by dividing the sub-message 302 (A(x)) by the composite CRC generating polynomial, P(x). Thus, in the second method 300, only one division is performed as the message is processed. However, the remainder, which, is typically much shorter than the message, is divided by the two factors of the CRC polynomial as described below. The division in step 304 is represented by the following Equation 8:

Rp(x)=A(x) mod P(x)  Equation 8

[0075] Rp(x) is the CRC of the sub-message, A(x), without trailing zeroes (i.e., without regard to position). Rather than directly adjusting Rp(x) using the optimal method as could be done if P(x) were primitive, in step 308 of the second method 300, the composite partial remainder 306 (Rp(x)) is divided by the first factor, P1(x), to obtain a first factor partial remainder 310 (R1p(x)). The division is step 308 is represented by the following Equation 9:

R1p(x)=Rp(x) mod P1(x)  Equation 9

[0076] In step 316, the composite partial remainder 306 (Rp(x)) is divided by the second factor, P2(x), to obtain a second factor partial remainder 318 (R2p(x)). The division in step 316 is represented by the following Equation 10:

R2p(x)=Rp(x) mod P2(x)  Equation 10

[0077] In step 312, the first factor partial remainder 310 (R1p(x)) is adjusted based on the number, n, of trailing zeros in the composite sub-message to generate an adjusted first factor remainder 314 (R1(x)). The adjustment in step 312 is represented by the above Equation 6.

[0078] Likewise, in step 320, the second factor partial remainder 318 (R2p(x)) is adjusted based on the number, n, of trailing zeros in the composite sub-message to generate an adjusted second factor remainder 322 (R2(x)). The adjustment in step 320 is represented by the above Equation 7.

[0079] In one embodiment, the adjustment performed in steps 312 and 320 is accelerated by using the optimal method described above, since the factors P1(x) and P2(x) are irreducible.

[0080] In step 324, an adjusted composite remainder or incremental CRC 326 (R(x)), which is an adjusted version of Rp(x), is obtained from the individual remainders R1(x) and R2(x) by a mapping described in further detail below.

[0081] In one form of the invention, method 300 is preferable over method 200 because the computations on the composite partial remainder 306 (Rp(x)) at steps 308 and 316 of method 300 are shorter than the computations on the sub-message 202 at steps 204 and 212 of method 200.

[0082] Method 200 (FIG. 10) and method 300 (FIG. 11) are simplified if one of the factors of P(x) is x+1. For instance, if one of the factors of P(x) is x+1, the division in step 212 or step 316 is trivial, as the computation is nothing more than a computation of parity. As an example, for the second method 300, if the second factor P2(x) is x+1, rather than using the optimal method to accelerate the adjustment, R2(x) is set equal to zero if Rp(x) has an even number of 1's, and R2(x) is set equal to one if Rp(x) has an odd number of 1's. If P2(x) is not x+1, R2p(x) is computed based on Equation 10 above in step 316, and the optimal method described above is used in one embodiment to compute R2(x)=R2p(x)*x^(n) mod P2(x) in step 320.

[0083] Computing R1p(x) is also simplified for the case where one of the factors of P(x) is x+1. If P2(x) is x+1, P1(x) will have an order of one less than P(x). Assuming “o” is the order of P(x), R1p(x) can be computed as follows: (1) Test the most significant bit of Rp(x); (2) if the most significant bit of Rp(x) is 0, then R1p(x) is the “o” minus one least significant bits of Rp(x); and (3) if the most significant bit of Rp(x) is 1, then R1p(x) is the “o” minus one least significant bits of the XOR of Rp(x) and P1(x).

[0084] For the case where P1(x) has an order of at least two less than the order of P(x) (i.e., P2(x) is not x+1), then normal mod 2 polynomial division is used to find R1p(x). This division can be done by processing one bit at a time. CRC methods that process multiple bits at the same time and do the mod operation in one step can also be used.

[0085] The mapping performed in step 220 (FIG. 10) and step 324 (FIG. 11) will now be described in further detail. In one embodiment, the mapping is a one to one mapping based on the Chinese Remainder Theorem, which is used to determine R(x) from the pair R1(x) and R2(x). In the following example, the fact that the message is a sub-message will be ignored, since a process of adjusting has been described above, and the issue addressed here is the mapping from individual remainders to a composite remainder. For that reason, the message will be called M(x) instead of A(x).

[0086] Assume, for example, that the composite CRC generating polynomial, P(x), is x⁴+x³+x²+1. The factors of this example polynomial, P(x), are P2(x)=x+1 and P1(x)=x³+x+1. The remainder of the division of the message M(x) by P1(x) is R1(x). Similarly the remainder of the division of the message M(x) by P2(x) is R2(x). In one embodiment, R1(x) and R2(x) are computed using the optimal method described above. Table 4 below shows how the remainder R(x) of the division of M(x) by P(x) is related to the individual remainders R1(x) and R2(x). TABLE 4 Output Input pattern pattern R1(x) R2(x) R(x) 000 0 0000 000 1 1011 001 0 1010 001 1 0001 010 0 1001 010 1 0010 011 0 0011 011 1 1000 100 0 1111 100 1 0100 101 0 0101 101 1 1110 110 0 0110 110 1 1101 111 0 1100 111 1 0111

[0087] Note that when a polynomial is represented with binary coefficients as a binary pattern, as in the Table 4 above, the binary coefficients are shown with the most significant bit on the left.

[0088] To determine the mapping, the system of modular equations given by the following Equations 11 and 12 is solved:

R1(x)=R(x) mod P1(x)  Equation 11

R2(x)=R(x) mod P2(x)  Equation 12

[0089] In Equations 11 and 12, P1(x) and P2(x) are the factors of the polynomial P(x), and R(x) is the remainder of the division of the message M(x) by P(x). We want to find the polynomial R(x) that satisfies the above two modular polynomial equations given in Equations 11 and 12. This can be pre-computed and is constant for a given CRC polynomial P(x). In one embodiment, the mapping from individual remainders R1(x) and R2(x) to a composite remainder R(x)is implemented using fixed combinational logic, since the mapping is known beforehand as it does not depend on the message M(x).

[0090] The four shaded entries in Table 4 (i.e., the second, third, fifth, and ninth entries) were generated by solving the modular system of equations given in Equations 11 and 12 above, using a commercially available mathematical computation software package. Note that for the four shaded entries, the input pattern, which includes R1(x) and R2(x), has a single “1”.

[0091] All entries in the non-shaded rows of Table 4 can be derived from those in the shaded rows as follows. Consider a non-shaded input pattern formed by concatenating the individual remainders R1(x) and R2(x). To find the output pattern R(x) corresponding to that non-shaded input pattern, the shaded input patterns that need to be XORed in order to produce the non-shaded input pattern are identified. The entries in the output pattern column, R(x), corresponding to the identified shaded input patterns are XORed, and the result of the XOR is the desired non-shaded output pattern.

[0092] For example, to find R(x) when R1(x) is 011 and R2(x) is 1 (i.e., the input pattern shown in the eighth row of Table 4), the following shaded output patterns are XORed: 1011, 1010, 1001. The result is 1000 which is the output pattern corresponding to the input pattern 0111.

[0093] Each bit b in R(x) is the XOR of the bits of R1(x) and R2(x) for each shaded entry in which bit b is a 1 in the output pattern. Assuming the bits of R1(x) are a1, b1, and c1, with al being the most significant bit; the bit of R2(x) is a2; and the bits of R(x) are a, b, c, and d, with a being the most significant bit; by examination of the Table 4 above, the values for a, b, c, and d may be determined as shown in the following Equations 13-16, respectively:

a=a1XOR b1XOR c1XOR a2  Equation 13

b=a1  Equation 14

c=a1XOR c1XOR a2  Equation 15

d=a1XOR b1XOR a2.  Equation 16

[0094]FIG. 12 is a schematic diagram illustrating a circuit 400 for implementing the mapping represented by Equations 13-16 above for the example CRC generating polynomial P(x)=x⁴+x³+x²+1. The three bits a1, b1, and c1, of the remainder 402 (R1), and the one bit a2 of the remainder 406 (R2), are provided as inputs to XOR gates 404A-404C (collectively referred to as XOR gates 404). The outputs of XOR gates 404 provide bits a, c, and d, of the adjusted composite remainder 408 (R), and the bit b of the adjusted composite remainder 408 (R) is the same as bit a1 of the remainder 402 (R1).

[0095] Although specific embodiments have been illustrated and described herein for purposes of description of the preferred embodiment, it will be appreciated by those of ordinary skill in the art that a wide variety of alternate and/or equivalent implementations may be substituted for the specific embodiments shown and described without departing from the scope of the present invention. Those with skill in the mechanical, electro-mechanical, electrical, and computer arts will readily appreciate that the present invention may be implemented in a very wide variety of embodiments. This application is intended to cover any adaptations or variations of the preferred embodiments discussed herein. Therefore, it is manifestly intended that this invention be limited only by the claims and the equivalents thereof. 

What is claimed is:
 1. A method of generating a CRC for a composite sub-message based on a CRC generating polynomial having at least two factors, the composite sub-message including sub-message data and a number, n, of trailing zeros, the method comprising: generating a first remainder based on the sub-message data and a first factor of the CRC generating polynomial; generating a second remainder based on the sub-message data and a second factor of the CRC generating polynomial; and generating the CRC for the composite sub-message based on adjusted versions of the first and the second remainders.
 2. The method of claim 1, and further comprising: adjusting at least one of the first and the second remainders based on the number, n, of trailing zeros in the composite sub-message.
 3. The method of claim 2, wherein the first remainder is an m-bit remainder, and wherein the adjusting step comprises: storing the first remainder in an m-bit memory location; examining each bit of N, where N equals n mod (2^(m)−1); and selectively advancing the contents of the m-bit memory location to a next state based on a value of each bit of N, the next state determined by a Galois field defined by the first factor.
 4. The method of claim 3, wherein the second remainder is adjusted in substantially the same manner as the first remainder.
 5. The method of claim 2, wherein the first remainder is an m-bit remainder, and wherein the adjusting step comprises: storing the first remainder in an m-bit memory location; and examining each bit of N, where N equals n mod (2^(m)−1), in order from a most significant bit to a least significant bit; the examining act for each examined bit comprising: finite field squaring the contents of the m-bit memory location, and; if the examined bit equals one, advancing the contents of the m-bit memory location to a next state as determined by a Galois field defined by the first factor.
 6. The method of claim 5, wherein the second remainder is adjusted in substantially the same manner as the first remainder.
 7. The method of claim 1, wherein the step of generating a first remainder comprises: dividing the sub-message data by the first factor.
 8. The method of claim 7, wherein the step of generating a second remainder comprises: dividing the sub-message data by the second factor.
 9. The method of claim 1, wherein the step of generating a first remainder comprises: dividing the sub-message data by the CRC generating polynomial, thereby generating an unadjusted composite remainder; and dividing the unadjusted composite remainder by the first factor, thereby generating the first remainder.
 10. The method of claim 9, wherein the step of generating a second remainder comprises: dividing the unadjusted composite remainder by the second factor, thereby generating the second remainder.
 11. The method of claim 1, wherein the step of generating the CRC comprises mapping the adjusted versions of the first and the second remainders to a corresponding CRC.
 12. The method of claim 11, wherein the mapping is performed with a fixed logic circuit.
 13. The method of claim 1, wherein the first and the second factors are primitive polynomials.
 14. The method of claim 1, wherein the first and the second factors are irreducible polynomials.
 15. A method of generating a CRC for a composite sub-message based on a CRC generating polynomial having at least two factors, the composite sub-message including sub-message data and a number, n, of trailing zeros, the method comprising: generating an unadjusted composite remainder representing a remainder of a division of the sub-message data by the CRC generating polynomial; generating a first factor remainder representing a remainder of a division of the unadjusted composite remainder by a first factor of the CRC generating polynomial; generating a second factor remainder representing a remainder of a division of the unadjusted composite remainder by a second factor of the CRC generating polynomial; and generating the CRC for the composite sub-message based on adjusted versions of the first factor remainder and the second factor remainder.
 16. The method of claim 15, and further comprising: generating the adjusted version of the first factor remainder based on the number, n, of trailing zeros in the composite sub-message using finite field squaring and advancing states in a Galois field defined by the first factor.
 17. The method of claim 15, wherein the second factor is x+1, the method further comprising: generating the adjusted version of the second factor remainder based on a parity computation of the unadjusted composite remainder.
 18. The method of claim 15, wherein the second factor is x+1, and wherein the CRC generating polynomial has an order of R, and wherein the step of generating a first factor remainder comprises: testing a most significant bit of the unadjusted composite remainder; setting the first factor remainder equal to the R−1 least significant bits of the unadjusted composite remainder if the tested bit is equal to zero; and setting the first factor remainder equal to the R−1 least significant bits of a result of an XOR of the unadjusted composite remainder and the first factor if the tested bit is equal to one.
 19. A method of generating a CRC for a composite sub-message based on a CRC generating polynomial having at least two factors, the composite sub-message including sub-message data and a number, n, of trailing zeros, the method comprising: generating a first factor remainder representing a remainder of a division of the sub-message data by a first factor of the CRC generating polynomial; generating a second factor remainder representing a remainder of a division of the sub-message data by a second factor of the CRC generating polynomial; and generating the CRC for the composite sub-message based on adjusted versions of the first factor remainder and the second factor remainder.
 20. The method of claim 19, and further comprising: adjusting at least one of the first factor remainder and the second factor remainder based on the number, n, of trailing zeros in the composite sub-message using finite field squaring and advancing states in at least one Galois field defined by at least one of the first and the second factors. 